LMFDB - Newform orbit 5175.2.a.ch

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5175,2,Mod(1,5175)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5175, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("5175.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$5175 = 3^{2} \cdot 5^{2} \cdot 23$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	5175.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$41.3225830460$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{7} - x^{6} - 13x^{5} + 11x^{4} + 33x^{3} - 9x^{2} - 13x - 1$ x^7 - x^6 - 13x^5 + 11x^4 + 33x^3 - 9x^2 - 13*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$2^{2}$
Twist minimal:	no (minimal twist has level 345)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$ -expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $q$ -expansion are expressed in terms of a basis $1,\beta_1,\ldots,\beta_{6}$ for the coefficient ring described below. We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + \beta_{2} q^{2} + ( - \beta_{3} + 2) q^{4} + (\beta_{5} + 1) q^{7} + (\beta_{4} + 2 \beta_{2} - \beta_1 + 1) q^{8} + (\beta_{6} + \beta_{5} + \beta_{4} + \cdots - 2) q^{11} - \beta_{4} q^{13} + ( - \beta_{6} + \beta_{4} + \beta_{2} + \cdots - 1) q^{14}+ \cdots + ( - 3 \beta_{4} - \beta_1 - 1) q^{98}+O(q^{100})$ q + b2 * q^2 + (-b3 + 2) * q^4 + (b5 + 1) * q^7 + (b4 + 2b2 - b1 + 1) q^8 + (b6 + b5 + b4 + b3 + 2b2 - 2) q^11 - b4 * q^13 + (-b6 + b4 + b2 - b1 - 1) * q^14 + (-b6 + b5 - 2b3 + 4) q^16 + (b1 + 2) * q^17 + (b3 + b2 + 2) * q^19 + (2b4 - b2 + b1 + 3) q^22 + q^23 + (-b6 - b5 - 2b4 - b3 - 2b2 + 2) * q^26 + (-b6 - 4b3 - 3b2 - b1 + 3) * q^28 + (-b4 + b3 + b2 + b1) * q^29 + (b6 + b5 + b2 - b1) * q^31 + (-b6 + b5 + b4 - b3 + 3b2 - 2b1) * q^32 + (2b6 + 2b4 + 3b3 + 5b2 - 2) * q^34 + (-b3 - 2b2 + 3) q^37 + (-b4 - b3 + b1 + 3) * q^38 + (-b4 - b3 - 3) * q^41 + (b6 + 2b4 + 3b2 - b1 + 1) * q^43 + (2b6 + 4b4 + 4b3 + 6b2 - 6) * q^44 + b2 * q^46 + (2b6 + 2b5 + b4 + 2b3 + 3b2 + b1 - 1) * q^47 + (-2b6 - 2b4 + b3 - b2 + b1 + 5) * q^49 + (-b6 - b5 - 2b4 - b3 - b2 - b1 - 1) q^52 + (2b6 - 2b4 - b2 - 3) * q^53 + (b5 - b3 + 5b2 - 3b1 - 3) * q^56 + (b6 - b5 - b4 + b3 - b2 + b1 + 3) * q^58 + (-b4 - 3b3 - b2 - b1 - 2) q^59 + (b6 + b5 - 3b4 + b3 - b2 + b1 + 1) q^61 + (-3b6 - b5 - b4 - 3b3 - 2b2 + 4) q^62 + (-2b6 - 5b3 - 3b2 - 3b1 + 7) * q^64 + (2b6 - b5 + 2b3 + b2 - b1 - 2) * q^67 + (2b6 + b4 - b3 - 2b2 + 3b1 + 7) q^68 + (-b6 + b5 - b4 + 3b2 + b1) q^71 + (-b6 - b5 + b4 + b3 + b2 + b1 + 3) * q^73 + (b4 + 2b3 + 5b2 - b1 - 7) * q^74 + (b6 - b5 + b4 + 4b2 - b1 - 3) q^76 + (-b6 - 3b5 - b4 + 3b3 + 2) * q^77 + (b6 + b5 + b4 - 2b2 + b1 + 1) q^79 + (-b6 - b5 - b4 - b3 - 3b2 - b1 + 3) q^82 + (3b4 - b3 + b2 - b1) q^83 + (b5 + 2b4 - 3b3 + 3b2 + b1 + 9) q^86 + (4b6 + 2b5 - 2b2 + 4b1 + 4) * q^88 + (b6 + 2b4 - 2b3 - 2) * q^89 + (-b6 + b5 + 3b4 + b3 + b2 - b1 + 1) q^91 + (-b3 + 2) * q^92 + (b6 - b5 + 4b4 + 3b3 + 2b2 + 2b1 + 2) * q^94 + (b6 + b5 + b3 + 2b2 + 2b1 + 4) * q^97 + (-3b4 - b1 - 1) q^98
$\operatorname{Tr}(f)(q)$	$=$	$7 q + 2 q^{2} + 14 q^{4} + 5 q^{7} + 12 q^{8} - 12 q^{11} + 2 q^{13} - 6 q^{14} + 24 q^{16} + 11 q^{17} + 16 q^{19} + 12 q^{22} + 7 q^{23} + 14 q^{26} + 16 q^{28} + q^{29} + 5 q^{31} + 6 q^{32} - 4 q^{34}+ \cdots + 2 q^{98}+O(q^{100})$ 7 * q + 2 * q^2 + 14 * q^4 + 5 * q^7 + 12 * q^8 - 12 * q^11 + 2 * q^13 - 6 * q^14 + 24 * q^16 + 11 * q^17 + 16 * q^19 + 12 * q^22 + 7 * q^23 + 14 * q^26 + 16 * q^28 + q^29 + 5 * q^31 + 6 * q^32 - 4 * q^34 + 17 * q^37 + 20 * q^38 - 19 * q^41 + 14 * q^43 - 34 * q^44 + 2 * q^46 - 6 * q^47 + 30 * q^49 - 2 * q^52 - 15 * q^53 - 4 * q^56 + 22 * q^58 - 11 * q^59 + 8 * q^61 + 22 * q^62 + 48 * q^64 - 3 * q^67 + 38 * q^68 + q^71 + 18 * q^73 - 38 * q^74 - 8 * q^76 + 20 * q^77 - 2 * q^79 + 20 * q^82 - q^83 + 60 * q^86 + 16 * q^88 - 16 * q^89 + 2 * q^91 + 14 * q^92 + 8 * q^94 + 26 * q^97 + 2 * q^98

Basis of coefficient ring in terms of a root $\nu$ of $x^{7} - x^{6} - 13x^{5} + 11x^{4} + 33x^{3} - 9x^{2} - 13x - 1$ x^7 - x^6 - 13*x^5 + 11*x^4 + 33*x^3 - 9*x^2 - 13*x - 1 :

$\beta_{1}$	$=$	$( -\nu^{6} + 4\nu^{5} + 11\nu^{4} - 34\nu^{3} - 11\nu^{2} - 18\nu - 3 ) / 20$ (-v^6 + 4v^5 + 11v^4 - 34v^3 - 11v^2 - 18*v - 3) / 20
$\beta_{2}$	$=$	$( -3\nu^{6} + 2\nu^{5} + 43\nu^{4} - 22\nu^{3} - 133\nu^{2} + 16\nu + 41 ) / 20$ (-3v^6 + 2v^5 + 43v^4 - 22v^3 - 133v^2 + 16v + 41) / 20
$\beta_{3}$	$=$	$( 4\nu^{6} - 6\nu^{5} - 49\nu^{4} + 66\nu^{3} + 94\nu^{2} - 58\nu - 3 ) / 10$ (4v^6 - 6v^5 - 49v^4 + 66v^3 + 94v^2 - 58v - 3) / 10
$\beta_{4}$	$=$	$( 9\nu^{6} - 6\nu^{5} - 119\nu^{4} + 66\nu^{3} + 319\nu^{2} - 28\nu - 113 ) / 20$ (9v^6 - 6v^5 - 119v^4 + 66v^3 + 319v^2 - 28v - 113) / 20
$\beta_{5}$	$=$	$( 6\nu^{6} - 9\nu^{5} - 76\nu^{4} + 104\nu^{3} + 176\nu^{2} - 147\nu - 62 ) / 10$ (6v^6 - 9v^5 - 76v^4 + 104v^3 + 176v^2 - 147v - 62) / 10
$\beta_{6}$	$=$	$( -7\nu^{6} + 8\nu^{5} + 87\nu^{4} - 88\nu^{3} - 187\nu^{2} + 84\nu + 39 ) / 10$ (-7v^6 + 8v^5 + 87v^4 - 88v^3 - 187v^2 + 84v + 39) / 10

Display $\nu^j$ in terms of $\beta_i$

$\nu$	$=$	$( \beta_{6} + \beta_{4} + \beta_{3} + \beta_{2} ) / 2$ (b6 + b4 + b3 + b2) / 2
$\nu^{2}$	$=$	$( \beta_{6} + \beta_{5} + 2\beta_{4} - \beta_{3} + 3\beta_{2} - \beta _1 + 7 ) / 2$ (b6 + b5 + 2b4 - b3 + 3b2 - b1 + 7) / 2
$\nu^{3}$	$=$	$( 8\beta_{6} + \beta_{5} + 7\beta_{4} + 8\beta_{3} + 8\beta_{2} + 3\beta _1 + 1 ) / 2$ (8b6 + b5 + 7b4 + 8b3 + 8b2 + 3*b1 + 1) / 2
$\nu^{4}$	$=$	$3\beta_{6} + 4\beta_{5} + 9\beta_{4} - 5\beta_{3} + 17\beta_{2} - 4\beta _1 + 27$ 3b6 + 4b5 + 9b4 - 5b3 + 17b2 - 4b1 + 27
$\nu^{5}$	$=$	$( 67\beta_{6} + 6\beta_{5} + 61\beta_{4} + 71\beta_{3} + 71\beta_{2} + 38\beta _1 + 2 ) / 2$ (67b6 + 6b5 + 61b4 + 71b3 + 71b2 + 38b1 + 2) / 2
$\nu^{6}$	$=$	$( 33\beta_{6} + 67\beta_{5} + 164\beta_{4} - 105\beta_{3} + 335\beta_{2} - 67\beta _1 + 485 ) / 2$ (33b6 + 67b5 + 164b4 - 105b3 + 335b2 - 67b1 + 485) / 2

Display $\beta_i$ in terms of $\nu^j$

Embeddings

For each embedding $\iota_m$ of the coefficient field, the values $\iota_m(a_n)$ are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label

\iota_m(\nu)

a_{2}

a_{3}

a_{4}

a_{5}

a_{6}

a_{7}

a_{8}

a_{9}

a_{10}

1.1

−1.35688
2.21373
0.802220
−0.516375
−0.0831195
3.07982
−3.13940

−2.63914

4.96506

3.29106

−7.82519

1.2

−1.40464

−0.0269736

1.58054

2.84718

1.3

−1.27208

−0.381820

−3.58355

3.02986

1.4

0.161527

−1.97391

5.15575

−0.641892

1.5

1.93829

1.75698

−3.86288

−0.471035

1.6

2.43902

3.94880

−0.839790

4.75315

1.7

2.77702

5.71186

3.25887

10.3079

Atkin-Lehner signs

$p$	Sign
$3$	$-1$
$5$	$-1$
$23$	$-1$

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	5175.2.a.ch		7
3.b	odd	2	1		1725.2.a.bk		7
5.b	even	2	1		5175.2.a.ca		7
5.c	odd	4	2		1035.2.b.f		14
15.d	odd	2	1		1725.2.a.bl		7
15.e	even	4	2		345.2.b.d	✓	14

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
345.2.b.d	✓	14	15.e	even	4	2
1035.2.b.f		14	5.c	odd	4	2
1725.2.a.bk		7	3.b	odd	2	1
1725.2.a.bl		7	15.d	odd	2	1
5175.2.a.ca		7	5.b	even	2	1
5175.2.a.ch		7	1.a	even	1	1	trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(5175))$ :

T_{2}^{7} - 2T_{2}^{6} - 12T_{2}^{5} + 20T_{2}^{4} + 43T_{2}^{3} - 44T_{2}^{2} - 56T_{2} + 10

T_{7}^{7} - 5T_{7}^{6} - 27T_{7}^{5} + 141T_{7}^{4} + 158T_{7}^{3} - 1068T_{7}^{2} + 296T_{7} + 1016

T_{11}^{7} + 12T_{11}^{6} + 8T_{11}^{5} - 338T_{11}^{4} - 864T_{11}^{3} + 1728T_{11}^{2} + 3840T_{11} - 5120

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{7} - 2 T^{6} + \cdots + 10$ T^7 - 2T^6 - 12T^5 + 20T^4 + 43T^3 - 44T^2 - 56T + 10
$3$	$T^{7}$ T^7
$5$	$T^{7}$ T^7
$7$	$T^{7} - 5 T^{6} + \cdots + 1016$ T^7 - 5T^6 - 27T^5 + 141T^4 + 158T^3 - 1068T^2 + 296T + 1016
$11$	$T^{7} + 12 T^{6} + \cdots - 5120$ T^7 + 12T^6 + 8T^5 - 338T^4 - 864T^3 + 1728T^2 + 3840T - 5120
$13$	$T^{7} - 2 T^{6} + \cdots + 128$ T^7 - 2T^6 - 26T^5 + 18T^4 + 144T^3 - 96T^2 - 192T + 128
$17$	$T^{7} - 11 T^{6} + \cdots - 440$ T^7 - 11T^6 - T^5 + 275T^4 - 334T^3 - 1564T^2 + 2768*T - 440
$19$	$T^{7} - 16 T^{6} + \cdots + 2000$ T^7 - 16T^6 + 82T^5 - 54T^4 - 900T^3 + 3248T^2 - 4336T + 2000
$23$	$(T - 1)^{7}$ (T - 1)^7
$29$	$T^{7} - T^{6} + \cdots - 10000$ T^7 - T^6 - 75T^5 - 71T^4 + 1580T^3 + 3760T^2 - 3200*T - 10000
$31$	$T^{7} - 5 T^{6} + \cdots + 128$ T^7 - 5T^6 - 81T^5 + 189T^4 + 1820T^3 + 1328T^2 - 960T + 128
$37$	$T^{7} - 17 T^{6} + \cdots - 16$ T^7 - 17T^6 + 67T^5 - 3T^4 - 260T^3 + 48T^2 + 224T - 16
$41$	$T^{7} + 19 T^{6} + \cdots + 6640$ T^7 + 19T^6 + 107T^5 + 27T^4 - 1320T^3 - 2104T^2 + 4080T + 6640
$43$	$T^{7} - 14 T^{6} + \cdots + 44320$ T^7 - 14T^6 - 76T^5 + 1268T^4 + 624T^3 - 27280T^2 + 26752T + 44320
$47$	$T^{7} + 6 T^{6} + \cdots - 110720$ T^7 + 6T^6 - 216T^5 - 926T^4 + 11888T^3 + 25984T^2 - 104768T - 110720
$53$	$T^{7} + 15 T^{6} + \cdots + 87560$ T^7 + 15T^6 - 163T^5 - 3313T^4 - 1782T^3 + 163284T^2 + 632640T + 87560
$59$	$T^{7} + 11 T^{6} + \cdots - 18320$ T^7 + 11T^6 - 183T^5 - 2295T^4 - 700T^3 + 33424T^2 - 4128T - 18320
$61$	$T^{7} - 8 T^{6} + \cdots + 3215456$ T^7 - 8T^6 - 282T^5 + 1986T^4 + 23240T^3 - 142944T^2 - 565184T + 3215456
$67$	$T^{7} + 3 T^{6} + \cdots - 653960$ T^7 + 3T^6 - 351T^5 - 1051T^4 + 31478T^3 + 99036T^2 - 418600T - 653960
$71$	$T^{7} - T^{6} + \cdots + 709040$ T^7 - T^6 - 265T^5 + 619T^4 + 19056T^3 - 63816T^2 - 205200*T + 709040
$73$	$T^{7} - 18 T^{6} + \cdots + 2048$ T^7 - 18T^6 - 18T^5 + 1402T^4 - 2320T^3 - 20736T^2 + 29696T + 2048
$79$	$T^{7} + 2 T^{6} + \cdots + 6400$ T^7 + 2T^6 - 288T^5 - 960T^4 + 12216T^3 + 1504T^2 - 65920T + 6400
$83$	$T^{7} + T^{6} + \cdots - 2560$ T^7 + T^6 - 235T^5 + 507T^4 + 12488T^3 - 46944T^2 - 28800*T - 2560
$89$	$T^{7} + 16 T^{6} + \cdots + 272960$ T^7 + 16T^6 - 168T^5 - 2744T^4 + 2936T^3 + 101776T^2 + 312288T + 272960
$97$	$T^{7} - 26 T^{6} + \cdots + 955456$ T^7 - 26T^6 + 48T^5 + 2700T^4 - 10464T^3 - 90848T^2 + 277824T + 955456
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